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SPECTRAL THEORY OF HUBERT       401
C 6 Sp(N), it follows from (1 s/| 1 8

ll(£ °/)(AO - (* °/)WII S e. Since that "^/W) " A(/W)II ^ e and that
result.                                               e was arbitrary, this gives the required

Finally, to prove (iv), note that h

bounded, by virtue of (15.11 .8.1 )3 it i ecause the sequence of norms \\fk(N)\\ is
sequence (fk(N) ' x) for x belongin Sen°u£h to prove the convergence of the
(7.5.5)). With the notation of (1S%11^vto a total subset of E ((12.15.7.1) and
proving the assertion for each Jv *>' We may therefore restrict ourselves to
multiplication by the class of th*" ut since fk(Nn) may be identified with

, the result follows fr ^^on of /* to SPW) in the sPace
*r

(15.11 .9)    For each universally      a

closed subspace ofE, stable with res™^1* subset M °/sPW» let E(M)
under the orthogonal projector p     Pect to N and JV*, which is the image of E
the restriction of N to E(M) fc c^} .^ ^M^ (15.10.6). Tfe/2 //ze spectrum of

tame<l in M (closure in C).
For each n, let EM(M) be the im

<PM(^»)- lt is immediately seen thaf^°f E" Under the orth°g°nal projector
hence (15.11.5) it is enough to prov    ^) « the Hilbert sum of the EM(M),
there exists a continuous function   ^ ^ Proposition for each Nn . If a ^ M
for all C 6 SpW.) (4.5.1). It f^^ Sp(AQ such that ^(Q(a - 0<?M(0 =

• AT*--,                 -        -             -           /*   J.L  lOll^x-,                                       «^\3/\-        3^-rmv^/

to En(M), then alEn(M) - Ni has an j       S that> if Nn is the restriction of A^
En(M).                                           averse equal to the restriction ofg(Nn) to

Remark

(15.11.10)    Note that the ^

for each positive measure ^ on c           Proves (ii) in (15.11.8) shows that,

by the class of lc in L|(K, M) i$ Wlth compact support K, multiplication
that Sp(N) = K (converse of (15 ^ 3^ n°rmal continuous operator N such

(15.11.11)    LetfbeahomeomorphiSn   f

subset N ofC containing Sp(jy) r/2e ^ °J a closed subset UofC onto a closed
operator N' on E whose spectrum * Te exists a unique normal continuous
f(N') = N.                                   S c°ntained in M and which is such that

If h is the homeomorphisrn Of >^

virtue of (15.11.8(iii)) we must hav0^^ ^ which is the inverse of/>then bY
versely the spectrum of A(N) is/z(Sp(^f' * *(/W) = W; and since con-
the result follows.                      ^ >> ^ M, it follows that/(/z(N)) = AT, andty of a. If J is infinite, then a